Splitting of ring extensions
Let R be an associative ring with identity. A right R-module P is called faithfully projective over R, if P is a projective R-module and if PM\(\neq P\) for each maximal two-sided ideal M of R. Cortzen, Small and Stafford have shown that if \(R\subseteq S\) is an extension of rings such that S is a projective right R-module, then the inclusion map splits if and only if S is faithfully projective [\textit{B. Cortzen, L. W. Small} and \textit{J. T. Stafford}, Proc. Am. Math. Soc. 82, 28-30 (1981; Zbl 0463.16017)]. The author shows that if R is assumed to be commutative, then the above result remains true if S is R-projective.
scientific article; zbMATH DE number 3950670